Adiabatic Berry Phase and Hannay Angle for Open Paths

Abstract

We obtain the adiabatic Berry phase by defining a generalised gauge potential whose line integral gives the phase holonomy for arbitrary evolutions of parameters. Keeping in mind that for classical integrable systems it is hardly clear how to obtain open-path Hannay angle, we establish a connection between the open-path Berry phase and Hannay angle by using the parametrised coherent state approach. Using the semiclassical wavefunction we analyse the open-path Berry phase and obtain the open-path Hannay angle. Further, by expressing the adiabatic Berry phase in terms of the commutator of instantaneous projectors with its differential and using Wigner representation of operators we obtain the Poisson bracket between distribution function and its differential. This enables us to talk about the classical limit of the phase holonomy which yields the angle holonomy for open-paths. An operational definition of Hannay angle is provided based on the idea of classical limit of quantum mechanical inner product. A probable application of the open-path Berry phase and Hannay angle to wave-packet revival phenomena is also pointed out.Comment: Latex, no figures, 24 pages, submitted to Ann. Phy

Phase-Space Approach to Berry Phases

“…Clearly, the corresponding canonical transformation should also eliminate the classical Hannay angles. Let us see how our basic formula (12) is related to the triviality problem analyzing once more a generalized harmonic oscillator (18). It was observed by Jackiw [17] and Sousa [14] that adding the total time derivative of '−Y q 2 /2X' to the classical Lagrangian one obtains canonically transformed…”

“…After this work was completed I have learned from A. K. Pati that he used Wigner-Weyl formalism [18] in deriving the classical limit of the adiabatic phase for noncyclic evolution.…”

Communication: Quantum dynamics in classical spin baths

“…[42][43][44] It is predicted by Eq. (6) to be present also for open paths 41 of the classical environment and it is off-diagonal in nature [45][46][47] since it can only be different from zero for the off-diagonal elements of ρ αα . It easy to see that the geometric phase φ αα can be obtained without invoking the adiabatic limit upon selecting the diagonal elements d I αα in Eq.…”

“…However, reasonable forms of the coupling between the quantum subsystem and the classical spin bath cause such a basis to be complex (whereas, when the bath is described by canonical variables, the adiabatic basis is real in the absence of magnetic fields). Hence, an interesting prediction of the quantum-classical Liouville equation for spin baths is the a) sergi@ukzn.ac.za presence of open-path 41 geometric phases [42][43][44] in the evolution of the off-diagonal elements [45][46][47] of the density matrix. In order to introduce the formalism, one can consider a classical spin vector S whose energy is described by the Hamiltonian H SB (S), and denote its components with S I , I = x, y, z.…”

Berry phase in superconducting charge qubits interacting with a cavity field

“…In this regard, the relation between entanglement and the Berry phase has been discussed in solid state systems [15] and in icosahedral Jahn-Teller systems [16]. Most of the earlier works on the geometrical phase focus on pure quantum states [17,18,19,20,21]. These types of systems, however, are very unrealistic and almost never occur in practice.…”